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I'm not sure if this type of question is acceptable here, but I'd really appreciate someone's help. I'm about to start writing a semestral work that we need to achieve the Bachelor's degree in our country. But my teacher is not an expert in this particular field so he told me I'll have to prepare it all mainly by myself. :-(

During the course of algebra, in part about UFD, PID and Euclidean domains, I found really interesting the structures $\mathbb{Z}[\sqrt{s}]$ where $s$ is a non-square integer. The properties of norm, irreducible and invertible elements etc. Also their classification as UFD, PID etc. is pretty interesting.
My questions are:

  1. What should be the topic of my work?
    I barely know the topic so I don't know if it can be somehow made in one whole topic - not just some chunk of some bigger topic with the unexpected end. I also can't decide whether is it enough / too big / too small topic - there are also other extensions of integers, there are also something called fields of rational numbers which my rings seem to be a part of...

  2. What would you recommend to be the content of my work?
    One chapter should be that basic introduction to $\mathbb{Z}[\sqrt{s}]$ - properties of the norm, invertible elements, algorithm for finding factorization to irreducible elements etc. Another one can be about classificaton of $\mathbb{Z}[\sqrt{s}]$ as UFD, PID, etc. Some facts seem to be easy - proof that $\mathbb{Z}[\sqrt{5}]$ is not a PID, $\mathbb{Z}[i]$ is Euclidean domain... But there are also some unsolved problems so I just want to be sure that I'm not digging into something way above my level.
    I can also focus on some particular ring, like $\mathbb{Z}[i]$, if there is something specially interesting on it.
    Any other chapters come into your mind?

  3. Literature? There is almost nothing in my language about it. In english I googled some classic books like G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers or D. S. Dummit, R. M. Foote: Abstract Algebra, both books slightly touching the topic but I would appreciate something (basic / advanced) more focused to study.

I guess here are many people with much better overview of the topic so I appreciate any comment / recommendation from you.

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Any book in algebraic number theory will surely cover these (rings/orders of integers of quadratic number fields). –  Jyrki Lahtonen Jun 26 '12 at 19:30
    
More focused literature would be algebraic number theory texts. –  anon Jun 26 '12 at 19:30
    
If your teacher/advisor is reasearch-active, a topic that the advisor chooses in consultation with you might give you the opportunity to experience research. –  André Nicolas Jun 26 '12 at 19:37
    
I can recommend Neukirch's Algebraic Number Theory. I haven't read the English translation but the German was used for reference in my courses. –  Cocopuffs Jun 26 '12 at 19:56

1 Answer 1

If $s$ is 1 more than a multiple of 4, we generally study ${\bf Z}[(1+\sqrt s)/2]$ instead of ${\bf Z}[\sqrt s]$. Find out why in the book by Stewart and Tall, or the book Number Fields by Marcus.

A couple of goals you might try to reach are the uniqueness of factorization into prime ideals in number fields, and the finiteness of the class number. You'll see these discussed in the books I mentioned, or any introductory text in Algebraic Number Theory.

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